DEPARTMENT OF GEOLOGY AND GEOPHYSICS, UNIVERSITY OF MINNESOTA, 310 PILLSBURY DRIVE SE, MINNEAPOLIS, MN , USA

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1 JOURNAL OF PETROLOGY VOLUME 40 NUMBER 5 PAGES Calculation of Peridotite Partial Melting from Thermodynamic Models of Minerals and Melts. III. Controls on Isobaric Melt Production and the Effect of Water on Melt Production M. M. HIRSCHMANN 1,4, P. D. ASIMOW 2,4, M. S. GHIORSO 3 AND E. M. STOLPER 4 1 DEPARTMENT OF GEOLOGY AND GEOPHYSICS, UNIVERSITY OF MINNESOTA, 310 PILLSBURY DRIVE SE, MINNEAPOLIS, MN , USA 2 LAMONT DOHERTY EARTH OBSERVATORY, PALISADES, NY 10964, USA 3 DEPARTMENT OF GEOLOGICAL SCIENCES, BOX , UNIVERSITY OF WASHINGTON, SEATTLE, WA 98195, USA 4 DIVISION OF GEOLOGICAL AND PLANETARY SCIENCES, CALTECH, PASADENA, CA 91125, USA RECEIVED APRIL 5, 1998; REVISED TYPESCRIPT ACCEPTED DECEMBER 2, 1998 We present a rigorous calculation of the isobaric entropy (S) change melting. This too can be demonstrated both by MELTS calculations of the melting reaction for peridotite ( S/ F ) P rxn, where F is the and by calculations in simple model systems. Productivities for melt fraction. Calculations at 1 and 2 GPa for fertile and depleted systems enriched in incompatible components are systematically peridotite show that ( S/ F ) P rxn varies as a function of extent of lower than those depleted in such components, though the total melt melting, temperature, and residual mineral assemblage. Changes in produced at any given temperature will be greater for an enriched reaction stoichiometry cause discontinuous changes in ( S/ F ) P rxn. system. Exhaustion of clinopyroxene from peridotite residua decreases Although calculated ( S/ F ) P rxn varies by about a factor of two calculated productivity by about a factor of four, and therefore (from ~0 25 to ~0 5 J/K per g), such variations have relatively extensive partial melting of harzburgitic residues is inhibited. Callittle effect on the formation of melt during adiabatic upwelling and culated isothermal addition of water to hot peridotite causes melting a characteristic value suitable for peridotite partial melting at least to increase roughly linearly with the abundance of water added to up to 3 GPa is 0 3 J/K per g. Calculated variations in isobaric the system, in agreement with the trend recognized earlier for melt productivity, ( F/ T ) P, are large and have a significant effect Mariana trough basalts. Melt production for calculated addition of on calculated adiabatic productivity, ( F/ P) S. For partial melting a subduction fluid (45 wt % H 2 O, 45% Na 2 O, 10% K 2 O) is of fertile peridotite, MELTS calculations suggest that near-solidus only slightly greater than for pure water. If water addition to productivities are greatly reduced relative to productivities at higher peridotite is not forced to be isothermal by an externally imposed melt fraction, owing to the incompatible behavior of Na 2 O and the heat sink or by buffering from low variance chemical reactions, then effect of this component on the liquidus temperature of partial melts. it will approach isenthalpic conditions, which will reduce melt This behavior can also be demonstrated in simple model systems. production per increment of water added by about a factor of two. Calculated near-solidus productivity for fractional or incremental For heating of peridotite containing minor amounts of H 2 O, batch melting of peridotite is lower than for batch melting, but after calculations suggest that the extent of melting will remain small a small amount of melting (~2%), productivity for the fractional (<5%) until the temperature is sufficient to generate significant or incremental batch melting case is greater than that of batch melt for an equivalent dry peridotite. Small degrees of melting deep Corresponding author. Present address: Department of Geology and Geophysics, University of Minnesota, 310 Pillsbury Drive SE, Minneapolis, MN , USA. Telephone: Fax: Marc.M.Hirschmann-1@umn.edu Oxford University Press 1999

2 JOURNAL OF PETROLOGY VOLUME 40 NUMBER 5 MAY 1999 in mantle source regions caused by alkalis, CO 2, and H 2 O probably methods for applying MELTS to peridotite partial melting result in several distinct melting regimes where melt productivities problems, compared MELTS calculations with periresult are very small and melt compositions are strongly influenced by high dotite partial melting experiments (especially at pressures concentrations of alkalis and/or volatiles. Such regions are almost near 1 GPa), and reviewed the strengths and limitations certainly in the garnet peridotite stability field, and owing to the of applications of the current MELTS calibration to small extents of melting and low productivities in these deep melting peridotite partial melting calculations. In this paper, we zones, they are likely regions for development of extreme U-series use MELTS to address the influence of key variables disequilibria. affecting melt production in peridotitic source regions: the distribution of entropy among liquid and solid phases during melting and the amount of melt produced as a function of changing temperature at constant pressure (the isobaric productivity ), emphasizing in particular the KEY WORDS: mantle melting; peridotite; hydrous melting; ridges; arcs effects of incompatible components (e.g. H 2 O, K 2 O, CO 2, etc.) on melt production at low melt fractions, and the amount of melt produced owing to fluxing of peridotite by H 2 O-rich fluids. INTRODUCTION For melting in response to adiabatic upwelling, a critical Partial melting of the mantle is one of the chief mechgenerated variable is the productivity, the amount of melt (F) anisms of energy and mass transfer between the Earth s per increment of upwelling. Productivity affects interior and the surface, and has been a topic of intensive the total volume of melt (and hence the thickness of study for decades. As a result, there are now reasonable crust) generated from ascending mantle with a given constraints on many aspects of mantle melting processes, entropy (or potential temperature; McKenzie, 1984). including the volumes of melt formed in various tectonic Also, variations in productivity influence the average environments, the temperatures and pressures prevailing depth of melting inferred from basalt geochemistry (Plank during melting, and the compositions of mantle source et al., 1995) and may exert important controls on melt regions. However, the energetics of mantle melting are segregation processes (Spiegelman, 1993; Asimow et al., only roughly understood and the quantitative effects of 1995). In many treatments of adiabatic upwelling, the volatile components are yet to be described adequately. productivity is assumed to be constant (Klein & Langmuir, Although understanding of mantle melting must be 1987; Niu & Batiza, 1991; Kinzler & Grove, 1992; grounded in high-quality phase equilibria experiments, Kinzler, 1997), but considerable uncertainty remains such experiments are not well suited for understanding about the appropriate value to use and recent theoretical the energetics of mantle melting or the effect of fluxing work (Asimow et al., 1997) has shown that it probably components such as H 2 O on melt production. As is the increases significantly as melting proceeds (until ex- case for understanding the relationship between source haustion of cpx from the residue). Two variables that composition, melting process, and the composition of must be known in order to calculate adiabatic melt mantle melts (Hirschmann et al., 1999), forward models productivity are the isobaric change in entropy (S) as- linking phase equilibria to mass and energy balance are sociated with the melting reaction, ( S/ F ) P rxn (Asimow, required. Here we apply the MELTS algorithm (Ghiorso 1997), and the isobaric melt productivity, ( F/ T ) P, & Sack, 1995) to the problem of isobaric melt production where F is the mass fraction of melt relative to the original in the shallow mantle. MELTS has particular potential starting mass of peridotite. Neither of these variables is to aid understanding of melting behavior because it well characterized for peridotite melt systems. Evaluation explicitly incorporates a quantitative description of meltof of these variables and what controls them is one focus ing energetics. Such energetics play key roles in natural this paper. processes yet cannot be inferred directly from experiments Present understanding of the isobaric entropy of the on complex systems. melting reaction, ( S/ F ) P rxn, is particularly problematic. In a companion paper (Hirschmann et al., 1998b), we Not only are the values of this variable poorly known, show that MELTS calculations capture the essential but just what it actually corresponds to and how it features of the phase equilibria of partially melting periunderstood. can be related to measurable quantities is not widely dotite up to the lowest pressures of the garnet stability Based on analogy with one-component sys- field. Although the calculations have inaccuracies, the tems, ( S/ F ) P rxn is commonly equated with the entropy extent of agreement with melting experiments on peri- of fusion (ΔS fus ) (McKenzie & Bickle, 1988; Scott, 1992; dotitic compositions is sufficient to allow numerical simcomponent Iwamori et al., 1995). This is a poor analogy for multi- ulations of mantle melting processes that lead to useful systems, for which the entropy of fusion is insights that are not otherwise available. In our earlier poorly defined. In one-component systems isobaric melt- paper (Hirschmann et al., 1998b), we described calculation ing takes place at a fixed temperature and involves 832

3 HIRSCHMANN et al. PERIDOTITE PARTIAL MELTING III conversion of a solid to a liquid of the same composition, peridotite in the mantle wedge depends approximately and the entropy of fusion is just the difference in entropy linearly on the amount of the slab-derived fluid and/ between the coexisting, constant composition solid and or melt added to the peridotite. Melting experiments liquid at this temperature and pressure. In contrast, performed on peridotite with small amounts of added isobaric melting of natural peridotite takes place over a water (Hirose & Kawamoto, 1995) are consistent with range of temperatures, and the liquid formed does not the relationship inferred by Stolper & Newman up to have the same composition as the solid residue. Thus, the exhaustion of cpx. However, there have so far been for peridotite melt systems, ( S/ F ) P rxn is affected by no phase equilibrium calculations presented that explore differences in liquid and solid composition and by variof this behavior or that describe the energetic consequences ations in reaction stoichiometry as melting proceeds. addition of incompatible elements (of which water is Therefore, evaluation of this term in an expression for a very important example) to partially molten peridotite. productivity requires detailed information about the spewater MELTS incorporates the effects of Na 2 O, K 2 O, and cific entropies of solid and liquid phases as well as about on the thermodynamics of partially molten silicate reaction stoichiometry, and simple notions of the entropy systems, and therefore allows evaluation of the effects of of fusion based on analogy with a one-component system such components on peridotite melting in the upper (e.g. the difference in specific entropy between the bulk mantle. solid peridotite and a liquid of the same composition; or the difference in specific entropy between coexisting solids and liquids) are not applicable. MELTS is well suited to calculation of ( S/ F ) P rxn during partial melting of natural THE ENERGETICS OF PERIDOTITE peridotite. Knowledge of the isobaric productivity, ( F/ T ) PARTIAL MELTING P,of peridotite is also incomplete and experimental studies If melting during mantle upwelling is adiabatic and have yielded conflicting information. Some experimental reversible, then it is an isentropic process and the melt studies suggest that productivity in peridotite systems is production for batch melting processes is given by nearly constant from near the solidus to the exhaustion of cpx (Baker & Stolper, 1994; Robinson et al., 1998), but others suggest that melting is eutectic-like in that F = P S significant melt is generated over a small temperature C p sol +F(C liq p C p sol ) T T P F Vsol α sol +F(V liq α liq V sol α sol )+ S X interval above the solidus ( Jaques & Green, 1980), and P F still other studies suggest that ( F/ T ) P is significantly smaller near the solidus than at higher melt fractions ( Mysen & Kushiro, 1977; Walter & Presnall, 1994). The divergence of experimental evidence on this matter illustrates the difficulty of carrying out experiments near the solidus and underscores that the causes of variations in ( F/ T ) P, which propagate into adiabatic productivity, remain poorly understood. MELTS calculations can be used to gain a fuller understanding of the expected behavior of ( F/ T ) P as melting proceeds and of expected variations related to melt removal or phase exhaustion. Another topic of current interest is the effect of incompatible components present at higher than trace levels (e.g. Na 2 O, K 2 O, H 2 O, CO 2 ) on peridotite melting. For example, melting in the mantle wedge above sub- duction zones is driven by addition of water-rich fluids or melts from the subducting slab (e.g. Gill, 1981; Tatsumi, 1983; Tatsumi et al., 1986; Kushiro, 1987; Plank & Langmuir, 1988; Davies & Stevenson, 1992; Stolper & Newman, 1994). From studies of back-arc basin basalts from the Mariana trough, Stolper & Newman (1994) inferred that the water-rich component that fluxes the mantle wedge is rich in Na 2 O and K 2 O. By comparing the budgets of trace elements and water in these magmas, they inferred that the extent of melting experienced by C sol p +F(C liq T F T P p C p sol ) rxn F P + S (Asimow et al., 1997). In this equation, C p, V, and α are heat capacity, molar volume, and thermal expansivity. ( T/ P ) F is the slope of a constant melt fraction isopleth and ( F/ T ) P is the isobaric productivity (discussed in greater detail below); both of these derivatives can be estimated from detailed phase equilibrium measure- ments or from accurate thermodynamic calculations. ( S/ F ) P rxn, the isobaric entropy of melt reaction, is given by F S rxn =S l S s+ P S X (2) F P (Asimow et al., 1997), where S l is the specific entropy of the liquid, S s the specific entropy of the bulk solid coexisting with that liquid, and ( S X / F) P is shorthand for the summation in equation (3), which accounts for the (1) 833

4 JOURNAL OF PETROLOGY VOLUME 40 NUMBER 5 MAY 1999 changes in the entropy of the system as a result of redistribution of components between phases: S X = n1 F P i=1 S l m l i T,P,m j=/i ml i l F P,m j=/1 + (3) l i=1 ns S S m S i T,P,m j=/i ms i s F P,m j=/i. s Here S lors m l lors i T,P,m j=/i is partial specific entropy of liquid or solid component i, mlors i lors F P,mj=/i Fig. 1. MELTS calculation of ( S/ F ) rxn P as a function of melt fraction for the fertile MM3 peridotite composition at 1 GPa and 2 GPa and for the depleted DMM1 peridotite composition at 1 GPa. The trend for is the extensive change in mass of each liquid or solid each calculation has three distinct regions, separated by discontinuous changes in ( S/ F ) rxn P that correspond to changes in the residual mineral component m lors i as melting proceeds (i.e. the extensive assemblage. Below ~18% melting for MM3 (7% for DMM1), the equivalent of the stoichiometric reaction coefficient) and residual assemblage is lherzolitic (ol + opx + cpx + sp). Above this n s is the sum of components in all the solid phases. melt fraction, the assemblage is harzburgitic (ol + opx ± sp). At higher melt fraction (~50% for DMM1, ~60 and ~80% for MM3 at 1 and It should be noted that in a one-component system, 2 GPa), opx is eliminated from the residual assemblage and the residual ( S/ F ) P rxn reduces to S l S s, which is just the entropy of assemblage is dunitic (ol ± sp). The sharp drops at >95% melting for fusion. The term ( S X / P ) F in the numerator of equation MM3 correspond to the exhaustion of olivine in the residue (leaving (1) represents the change in entropy owing to mineral only spinel). In the inset, the same data are plotted against temperature. This plot shows that when the residual assemblage is harzburgitic or reactions other than melting that occur when pressure dunitic, but not lherzolitic, ( S/ F ) rxn P depends mainly on temperature. changes. It is defined in the same way as ( S X / F) P in In the lherzolitic region, ( S/ F ) rxn P is more strongly influenced by melt equation (3), except that P is a variable and F is held fraction. Regions at very high temperature that have low values of ( S/ F ) rxn P correspond to calculated stability of a tiny amount of chromite constant. This term can be important when mineral near the liquidus. reactions, such as those associated with the garnet peridotite to spinel peridotite or spinel peridotite to plagioclase peridotite, occur (Asimow et al., 1995), but otherwise is of less importance than the other terms in equation (1). Calculation of ( S/ F ) P rxn requires (1) an inventory of the specific entropy of each component in each phase and (2) detailed knowledge of the proportions and com- positions of coexisting phases as a function of melt fraction. At this time, MELTS is the only available model for calculating formally and rigorously (subject to its assumed thermochemical models of the various phases involved) the compositions and specific entropies of all phases along a path through the peridotite melting in- terval. Also, unlike other available models, MELTS incorporates estimates of the entropies of mixing of all phases. Although subject to uncertainties, MELTS thus allows us to explore the magnitude and variations of this term affecting adiabatic melting. Using MELTS and equations (2) and (3), we have calculated ( S/ F ) P rxn for the fertile MM3 peridotite (Baker & Stolper, 1994) at 1 GPa from the solidus to the liquidus (Fig. 1). Except near the solidus, where ( S/ F ) P rxn approaches 0 5 J/K per g, calculated values of ( S/ F ) P rxn are <0 4 J/K per g, and between 3% and 50% melting they are in the narrow range of 0 3 ± 0 05 J/K per g. In detail, from the solidus to the liquidus, there are three distinct intervals in which ( S/ F ) P rxn (heavy solid curve) varies continuously. These regions of continuous variations are separated by discontinuities corresponding to the exhaustion of cpx and opx from the residue, so values of ( S/ F ) P rxn are distinct for systems with lherzolitic, harzburgitic, and dunitic residues. Such changes in ( S/ F ) P rxn owing to changes in the mineralogy of the residue are related to discontinuous changes in reaction stoichiometry [i.e. to changes in ( S X / F) P ; equation (3)]. Of the three regions, ( S/ F ) P rxn is lowest for harzburgitic residues and higher for dunitic and lherzolitic residues. Gradual increases in ( S/ F ) P rxn in the harzburgite and dunite regions reflect primarily rising temperature, as the higher heat capacity in silicate liquid relative to coexisting minerals results in the specific entropy of liquid increasing relative to solids with rising temperature. Significant variations in ( S/ F ) P rxn are predicted in the near-solidus lherzolitic region. 834

5 HIRSCHMANN et al. PERIDOTITE PARTIAL MELTING III to vary continuously with F] with values calculated in the same way except with ( S/ F ) P rxn held at some ar- bitrary constant value. Values of ( S/ F ) P rxn (or actually of its less-rigorous equivalent ΔS fus ) used in the literature range from 0 25 J/K per g (McKenzie & Bickle, 1988; Scott, 1992) to 0 4 J/K per g (McKenzie & O Nions, 1991). Recently, Kojitani & Akaogi (1997) estimated ( S/ F ) P rxn 0 40 ± 0 03 J/K per g for peridotite partial melting from combined 1 atm calorimetric de- terminations of melting in a simple CMAS analogue system and estimated corrections for the effect of more complex compositions, pressure, and reaction stoi- chiometry. In Fig. 2, we compare the instantaneous adiabatic productivity calculated by MELTS [i.e. with variable ( S/ F ) P rxn ] with that calculated with ( S/ F ) P rxn set to constant values of 0 25, 0 30 and 0 40 J/K per g for the case of adiabatic batch melting of MM3 peridotite at 1 GPa at temperatures ranging from the solidus to the liquidus. The calculated productivity shown in Fig. 2 at any given melt fraction thus corresponds to the in- stantaneous productivity that would be applicable for batch melting along an adiabatic path that has that melt fraction at 1 GPa. As is evident from Fig. 2, adiabatic productivity calculated with ( S/ F ) P rxn set to 0 30 J/K per g is most similar to that calculated with variable ( S/ F ) P rxn. Near the solidus, where ( S/ F ) P rxn varies most, adiabatic productivity is more sensitive to the larger variations in ( F/ T ) P [also in the denominator in equation (1)] that are also occurring [see below and Asimow et al. (1997)], so ( S/ F ) P rxn variations have relatively little effect. The correspondence of the full MELTS calculation to the ( S/ F ) P rxn = 0 30 J/K per g approximation is particularly good for adiabats that at 1 GPa are between the solidus and the exhaustion of orthopyroxene at ~60% melting. Other constant values for ( S/ F ) P rxn reproduce the variable ( S/ F ) P rxn trend in productivity less ac- curately, although they do not affect the overall shape of the productivity function. Our calculations show that the MELTS-predicted de- viations from constant values of ( S/ F ) P rxn depicted in Fig. 1 have relatively little effect on the calculated adia- batic productivity at 1 GPa. Although not shown, values near 0 30 J/K per g also reproduce calculated ( F/ P ) S trends for pressures at least up to 3 GPa and for other peridotite compositions, particularly between the solidus and exhaustion of cpx. We conclude that for most situ- ations, 0 30 J/K per g is an appropriate constant value for calculation of adiabatic melting of peridotite in the shallow mantle up to the point of opx exhaustion. ( S/ F ) P rxn also appears in the expression for productivity during incrementally isentropic fractional fusion; the val- ues quoted here for batch melting reactions are also likely to be appropriate for fractional fusion because the effect of the Na 2 SiO 3 component will again be most pronounced near the solidus, where variations in ( S/ F ) P rxn are least In Fig. 1, we compare the trend calculated for MM3 at 1 GPa with that calculated at 2 GPa and with that calculated for depleted DMM1 peridotite (Wasylenki et al., 1996; Hirschmann et al., 1998b) at 1 GPa. The trend of ( S/ F ) P rxn as a function of melt fraction is similar for all three calculations. In all cases, there are distinct regions corresponding to the different residual mineral assemblages, with the lowest values occurring for harzburgite assemblages, and highest values occurring near the solidus. Differences in absolute value reflect primarily the different temperature melt fraction trajectories for the different cases. Thus, if the three calculations are compared as a function of temperature (see inset to Fig. 1), they differ little in the harzburgite and dunite regions for the different compositions when these overlap in temperature. In contrast, variations in ( S/ F ) P rxn in the lherzolite region are variable among the three com- positions; these variations are related primarily to changes in melt composition and not to temperature, as large changes in ( S/ F ) P rxn are predicted in the near-solidus region over small temperature intervals. The large calculated changes in ( S/ F ) P rxn in the lherzolite region are related primarily to variations in the Na 2 SiO 3 component in the melt in this region (Hirsch- mann et al., 1998b). The partial specific entropy of Na 2 - SiO 3 liquid, S l in equation (3) m l Na 2SiO 3 P.,T,mj=/Na l 2SiO 3 is 3 2 J/K per g at 1400 C and 1 GPa. This is significantly higher than that of other components calculated in the MELTS model, which are all between 2 6 and 2 7 J/K per g at 1400 C and 1 GPa, except for Fe 2 SiO 4 (1 9 J/ K per g) and Mg 2 SiO 4 (2 85 J/K per g), so the increase in the concentration of Na 2 O as the solidus is approached translates directly into an increase in ( S/ F ) P rxn. It should be noted that because MELTS calculations exaggerate the concentration of Na 2 O in near-solidus liquids (Hirschmann et al., 1998b), the large calculated increases in ( S/ F ) P rxn near the solidus shown in Fig. 1 are also probably exaggerated. Although calculated variations in ( S/ F ) P rxn through the melting interval of peridotite are rather large, equa- tion (1) shows that adiabatic productivity is influenced by many variables. Therefore large variations in ( S/ F ) P rxn do not necessarily result in large variations in adiabatic productivity. Inspection of equation (1) shows that it is not the absolute magnitude of changes in ( S/ F ) P rxn that determines how ( F/ P ) S varies, but the magnitude of those changes relative to other terms in the denominator. One way to gauge the impact of variations in ( S/ F ) P rxn on ( F/ P ) S is to compare values of ( F/ P ) S calculated rigorously by MELTS [i.e. allowing all variables in the right-hand side of equation (1) 835

6 JOURNAL OF PETROLOGY VOLUME 40 NUMBER 5 MAY 1999 inferred by analogy with a rigorous thermodynamic analysis in simple systems (Asimow et al., 1997). It is commonly assumed that fractional melting is less productive than batch melting (Langmuir et al., 1992; Iwamori et al., 1995). The intuitive basis for this assumption in multicomponent systems is that relative to residues generated by batch melting, residues of fractional melting are more depleted in easily fusible components, so melting of fractional residues is expected to require higher temperatures. Although isobaric experiments (Hirose & Kawamura, 1994) and MELTS calculations (Hirschmann et al., 1998b) suggest that fractional (or incremental batch) melting of the same bulk composition is indeed less productive than batch melting during the first few percent of melting, both suggest that isobaric productivities for fractional fusion can actually be comparable with or greater than those for batch fusion during later increments of melting before the exhaustion of cpx from the residue. Fig. 2. Calculated adiabatic productivity, ( F/ P ) S (in units of percent melting per GPa) plotted against extent of melting (in percent) at 1 In this section, we analyze the principal factors in- GPa for the fertile MM3 peridotite composition. Higher extents of fluencing the variation in ( F/ T ) p for batch and fracmelting correspond to adiabats with higher potential temperatures (and tional melting, to provide insights into how productivity entropies), which therefore have achieved higher extents of melting is likely to vary in complex peridotitic systems and how when they traverse the depth at which the pressure is 1 GPa. Calit may be affected by differences in process (batch vs culations for the curve marked variable are performed for the MM3 composition with values of all parameters on the right-hand side of fractional melting) and in source composition (enriched equation (1) calculated with MELTS. Curves marked with numbers vs depleted peridotite). We begin by examination of a are calculated in the same manner, except that the value used for ( S/ F ) simple two-component system. Variations in productivity rxn P is held at a constant value (0 25, 0 3, and 0 4 J/K per g), rather than the value calculated with MELTS. The similarity between in this simple system show a number of interesting features the variable line and that calculated with ( S/ F ) rxn P held at 0 3 J/K relevant to understanding natural melting processes that per g should be noted. are common to more complex systems, including peridotite. Because the simple system is more amenable to important. A higher value of ( S/ F ) rxn quantitative analysis, we examine these features in detail. P applies after opx exhaustion, and this may be of importance to melting or We then examine a slightly more complex ternary system melt rock reactions associated with dunite formation (e.g. that mimics the behavior of peridotite in a semi-quan- Kelemen et al., 1995). titative fashion. Finally, we use MELTS to explore variations in ( F/ T ) p in model peridotite systems. Because MELTS cannot calculate true fractional melting processes, we instead calculate incremental batch melting FACTORS INFLUENCING ISOBARIC and removal with a small step size (0 1 vol. %). A common MELT PRODUCTIVITY theme in all three treatments is that variations in isobaric The isobaric melt productivity, defined as the change in melt productivity can be understood in terms of changes melt fraction with temperature at constant pressure, ( F/ in melt composition and that productivity variations T ) p, enters explicitly into the expression for isentropic are strongly influenced by the behavior of incompatible productivity [see equation (1)] and thus has a critical components present in modest abundances (in peridotite, influence on the amount of melt generated from a given these may include Na 2 O, K 2 O, H 2 O, CO 2,P 2 O 5, and source region during adiabatic upwelling ( McKenzie, TiO 2 ). Our treatment differs from that of Asimow et al. 1984; Miller et al., 1991; Langmuir et al., 1992; Asimow (1997), which emphasized changes in solid compositions et al., 1997). MELTS calculations suggest that for melting and which, aalthough rigorously correct, does not lend of fertile spinel peridotite ( F/ T ) p is highly variable itself as easily to an intuitive understanding of variations and is affected by phase exhaustion, melt removal, and in ( F/ T ) p. whether melting takes place near the solidus or at higher melt fractions (Hirschmann et al., 1998b). Through equation (1), these variations in ( F/ T ) p are largely responsible for the calculated variations in ( F/ P ) S shown We first explore a two-component system composed Two-component system in Fig. 2. The importance of these variations can also be predominantly of one component Z that forms a nearly 836

7 HIRSCHMANN et al. PERIDOTITE PARTIAL MELTING III pure solid phase and a trace amount of a second component Y that is incompatible in that solid phase. The liquidus temperature, T, in this system can be estimated by a simple approximation of the freezing point depression relation X liq Y =k(t fus Z T ) (4) (2) Systems with higher concentrations of the in- compatible component Y have lower isobaric productivities at a given F for both batch and fractional fus ΔH Z k= (5) RT Z fus2 processes (i.e. compare the two sets of curves in Fig. 3, for X bulk Y = or 0 002). This is contrary to what where ΔH Z fus is the enthalpy of fusion of pure solid Z, might be assumed based on intuition; namely, that fertile and R is the gas constant. It should be noted that the sources (i.e. those rich in incompatible, easily fusible freezing point depression (and consequently all the effects components) are usually thought to have higher prodescribed below) are, at the level of the approximation ductivities than depleted sources. However, the reason given by equation (4), independent of the identity of for the actual behavior is easy to see by inspection of the incompatible element; i.e. on a molar basis, all equations (8) and (9): enrichments in Y cause decreases incompatible elements have the same effect in the limit in df/dt because X bulk Y is in the denominators of these of zero concentration, and their effects are additive, such equations. that X liq Y in equation (4) can signify the molar sum of all (3) Productivity increases with increasing melt fraction incompatible elements. (Fig. 3) for both equilibrium and fractional fusion. The For batch melting rate of increase is related to the change in concentration X bulk of the incompatible component Y, as can be seen from X liq Y =k(t Z fus Y T )= (6) (F+(1 F )D sol/liq Y ) differentiation of equation (4) with respect to F: and for fractional melting F = k (11) T P liq X Y X liq Y =k(t Z fus T )= X bulk Y (1 F ) (1/D sol/liq 1) (7) D sol/liq F Y P where X bulk Y and D sol/liq Y are the molar concentration in which shows that df/dt is inversely proportional to the the bulk system and the molar partition coefficient, change in concentration of component Y with changing respectively, for the minor incompatible component Y F. Thus, productivity is small when the concentration in (Shaw, 1970). Equations (4), (6), and (7) can be differ- the liquid of Y in the melt changes rapidly (as it does for entiated and solved for df/dt: both batch and fractional fusion near the solidus). With increasing F, productivity increases as the concentration of Y decreases less rapidly, and levels off at higher F. Also, T F batch sol/liq k[(1 D Y )F+D sol/liq Y ] 2 = (8) P X bulk Y (1 D sol/liq Y ) because changes in Y are initially more rapid for fractional fusion, near the solidus fractional fusion will be less productive than batch fusion (but right at the solidus, kd sol/liq2 Y the productivities for the two processes are identical). This (Denbigh, 1981, p. 261), where X liq Y is the mole fraction of component Y in the liquid, T Z fus is the melting temperature of pure solid Z, and k is a constant given by (1) As F 0 (i.e. as the solidus is approached), the curves for batch and fractional fusion of the same initial bulk composition have the same productivity. This is seen graphically in Fig. 3 and can be shown quantitatively, because as at F 0, equations (8) and (9) are equal: T F batch = P T F fractional = kd sol/liq2 Y P X bulk Y (1 D sol/liq Y ). (10) F = T P X bulk Y (1 D sol/liq Y )(1 F ). (9) (1/D sol/liq 2) Y can be seen analytically by differentiation of equations (8) and (9), which gives that as F 0, To illustrate the effects of incompatible elements on isobaric productivity, we use these expressions to cal- F T batch =2 kd sol/liq Y (12) culate ( F/ T ) P X bulk P for progressive batch and fractional Y melting for a system where Y is incompatible (D sol/liq Y = 0 01), ΔH Z fus = 50 kj/mol and T Z fus = 1600 K. To and examine the effect of the variable concentration of the incompatible component, we calculate both batch and F T fractional =(1 D sol/liq Y ) kd sol/liq Y (13) fractional melting for two values of X bulk Y, and P X bulk Y The results are plotted in Fig. 3. Key features of the resulting trends include: Inspection of the equations shows that, from the identical 837

8 JOURNAL OF PETROLOGY VOLUME 40 NUMBER 5 MAY 1999 nearly pure and its melting is approximated as congruent. Treatment of the solid as a pure, congruently melting compound leads to several inadequacies for understanding natural systems with increasing melt fraction. For example, as melting proceeds and the concentration of the Y component in the melt decreases in the melt, productivity must tend towards very high values, and for fractional melting, it tends toward infinity. To enhance our insight into productivity variations, we now examine a ternary system for which productivity is not required to go to arbitrarily high values. Three-component system Fig. 3. Calculated isobaric productivity, ( F/ T ) P, for binary model The model ternary system predominantly consists of two described in text, consisting predominantly of one component (Z) but also containing a minor quantity of a component (Y) that is highly components, A and B, which form a solid solution, and incompatible in the solid phase. Calculations are given for batch (solid a small quantity of a third component, C, which is curves) and fractional (dashed curves) melting of systems with or incompatible in the solid solution (Fig. 4). This ternary of the Y component in the initial bulk system. It should be noted system more closely resembles peridotite than does the that for a given composition, fractional and batch melting produce the same productivity at the solidus, fractional melting is less productive binary described above because there is a slowly varying than batch melting near the solidus, and the opposite is true at higher background productivity, governed by the A B solid melt fractions; also, productivity is higher for the bulk system poor in solution loop (Fig. 4a), which is perturbed near the solidus the incompatible component. by the effect of the small quantity of the incompatible component C. Component C is therefore a proxy for values at the solidus, the productivity for batch melting Na 2 O in fertile peridotite or H 2 O, K 2 O, P 2 O 5, etc. in a initially (i.e. at very small F) increases more rapidly than damp and/or metasomatized peridotite. As we shall for fractional fusion because 2 > D sol/liq see, this system has the minimum complexity necessary Y. This yields the expected result that fractional fusion is less productive to mimic the essential features of productivity during than batch fusion. However, in this simple system, this isobaric melting of peridotite in the absence of solid solid result is valid only near the solidus: as melting proceeds, phase changes or exhaustion of a phase from the residue. absolute changes in the concentration of Y become We assume that partitioning of A and B between liquid smaller for fractional than for batch fusion (because the and solid is characterized by a single equilibrium constant, concentration in the liquid for fractional fusion is nearly K D, zero after a few percent melting), so fractional fusion K D = X liq A X sol B becomes more productive than batch fusion as the melt (16) X sol A X liq B fraction builds up (Fig. 3). The crossover between batch and fractional productivities is the condition and that partitioning of component C is governed by Henry s Law, T F batch = P T F fractional (14) P D sol/liq C = X sol C (17) which occurs when D sol/liq Y (1 F ) (1/D sol/liq Y sol/liq 2) =[(1 D Y )F+D sol/liq in other words, it depends only on D sol/liq Y. Although we shall see below that this model binary system illustrates many effects that are of general relevance to productivity variations of partially melting multicomponent system and thus allows them to be understood quantitatively and simply, it is unlike natural systems in important respects. To begin with, the residue is monomineralic so the effects of phase exhaustion on productivity are not included (see below). Also, the solid is X liq C Heat capacities of the liquid and solid are assumed to be equal. Under these conditions, the liquidus tem- Y ] 2 (15) perature, T for any liquid can be approximated using the cryoscopic equation (e.g. Carmichael et al., 1974, pp ), T= R fus ΔH A RT A fus ΔH fus A ln X liq A X A. (18) sol This approximation neglects the effects of the melting temperatures and ΔH fus of the B and C components, but provides an adequate description of a system in which 838

9 HIRSCHMANN et al. PERIDOTITE PARTIAL MELTING III Fig. 4. Calculated phase relations for the simple model ternary described in text. (a) Calculated phase relations along the A B join are those of a solid solution in equilibrium with an ideal partial melt. (b) The A-rich portion of the model ternary system with calculated isotherms and the calculated batch partial melt composition. The liquidus phase everywhere is the A B solid solution, with trace quantities of dissolved C. Numbers along the trace of the partial melt curve are percent melt present. It should be noted that the distance in temperature between constant increments of melt decreases as the total amount of melt increases, illustrating that ( F/ T ) P is small near the solidus and in- creases as melting proceeds. indicates bulk system composition. Fig. 5. Calculations of (a) isobaric productivity, ( F/ T ) P, (b) the concentration of incompatible component C in the liquid, and (c) temperature vs melt fraction (in percent) for the simple model ternary system described in the text and shown in Fig. 4. For each panel, calculations were performed for an initial bulk composition of 89 7 mol % A, 10% B and 0 3% C (curves labeled 1 and 3 ) or 89 9% A, 10% B and 0 1% C (curves labeled 2 ). Curves labeled 1 and 2 are for batch melting: curves labeled 3 are for incremental batch melting, with melting steps of 0 1%. the liquid is rich in the A component, and it is thus sufficient for our purposes. We assume values of K D, D C sol/liq, ΔH A fus and T A fus of 0 3, 0 01, 50 kj/mol, and 1600 K and a bulk composition of 89 7% A, 10% B, and 0 3% C. Given these values, the relationship between liquid composition, melt fraction, and temperature is completely described by equations (16) (18). The cal- culated phase relations along the A B join are shown in Fig. 4a, and the composition of liquids and position of isotherms in the ternary are shown in Fig. 4b. The relationships between temperature, melt composition, and melt fraction for batch and fractional fusion are illustrated by the curves labeled 1 and 3 in Fig. 5. The calculated productivity for this ternary solution shows the same essential features as the binary solution already discussed. Comparing batch vs fractional melting, productivity is the same at the solidus, higher for batch melting near the solidus, and higher for fractional melting above a certain critical degree of melting (~2 5% melting for this example). As in the case of the binary model, the key variable is the rate of change of the liquid composition as melt fraction increases. Reflecting the phenomenon of freezing point depression, C-rich liquids formed near the solidus have low melting temperatures (Fig. 5c). However, after only small increases in melt fraction, X C in the liquid is sharply reduced (simply because of the incompatibility of component C; Fig. 5b), and therefore the freezing point depression decreases substantially (i.e. the liquidus temperature rises sub- stantially; Fig. 5c). Decreases in the B content of the liquid contribute much less to liquidus temperature changes near the solidus, and become important relative 839

10 JOURNAL OF PETROLOGY VOLUME 40 NUMBER 5 MAY 1999 to the effects of C only as C in the liquid becomes more dilute at higher melt fractions. As implied at the end of the previous section on the binary model system, an important difference is that in the ternary system the productivity does not increase to very large values, but instead levels off [i.e. ( 2 F/ T 2 ) P is negative rather than positive above a few percent melting]; this is particularly apparent for the fractional fusion example, where the productivity becomes nearly constant at the background level of the A B binary once the concentration of component C becomes negligible above ~7 8% melting (as a result of which, batch fusion again becomes more productive than fractional fusion at above 16 17% melting). Thus, the generalizations in the previous section regarding productivity variations near the solidus and their causes are not artifacts of the approximation that the solid is a pure, congruently melting compound. Although the calculated productivity at any given F is always lower for enriched sources than for depleted sources (compare curves 1 and 2 in Fig. 5a) and fractional melting is more productive than batch melting over much of the melting interval, it is important to note that the total extent of melting at any given temperature is always greatest for batch melting of the enriched source for this simple system (i.e. above the solidus, curve 1 is further to the right than the others in Fig. 5c). Thus, the total amount of melt resulting from isobaric heating is greatest for more enriched sources and for batch melting. For the comparison between enriched and depleted sources, this is because enriched sources begin to melt at a lower temperature. For the comparison between batch and fractional processes, this is because fractional melt production never catches up with batch melt production, owing to its lower productivity near the solidus. These features are also seen in peridotite melting calculations, as discussed in the next section. Isobaric melt productivity near the peridotite solidus We now turn to MELTS calculations of isobaric productivity of fertile peridotite at 1 GPa. Calculations predict that ( F/ T ) p of fertile peridotite is small near the solidus and increases as melting proceeds up to the exhaustion of cpx from the residue (Hirschmann et al., 1998b). This is illustrated by curve 1 in Fig. 6a, which shows ( F/ T ) p calculated for batch melting of MM3 peridotite at 1 GPa from the solidus up to 20% melting. The calculated ( F/ T ) p increases from 0 01%/ C at the solidus to 0 6%/ C just before the exhaustion of cpx at ~18% melting. By analogy to the simple systems above, this dramatic increase in isobaric productivity reflects primarily the significant decrease in the concentration of Na in the liquid as melting proceeds with rising temperature (see curve 1 in Fig. 6b). We note again that Fig. 6. MELTS calculations of (a) isobaric productivity, ( F/ T ) P, (b) concentration of Na 2 O in the liquid, and (C) temperature vs melt fraction (in percent). In each panel, calculations are performed for batch melting of the fertile MM3 peridotite composition (curves labeled 1 ), batch melting of the depleted DMM1 peridotite composition (curves labeled 2 ), and incremental batch melting (i.e. an approximation to fractional fusion) of MM3, with melting steps of 0 1% (curves labeled 3 ). For incremental batch melting, the productivity is calculated relative to the original source mass (Asimow et al., 1997). Discontinuities in ( F/ T ) P at ~18% melt (MM3) and ~8% melt (DMM1) reflect exhaustion of cpx from the solid residue. Also shown in (a) are the isobaric batch and fractional productivities calculated from the models of Langmuir et al. (1992) (L 92) and Iwamori et al. (1995) (I 95). All calculations at 1 GPa, except the trend in (a) from Iwamori, which is at 1 5 GPa. although the change in ( F/ T ) p calculated by MELTS for the MM3 composition is exaggerated because the concentration of Na 2 O in calculated near-solidus liquids is too high (Hirschmann et al., 1998b), the overall form of the effect is robust. Because the decrease in isobaric productivity near the solidus is related to the change in the abundance of incompatible components in the melt, all other things being equal, we would predict that productivities for systems depleted in incompatible elements will be larger 840

11 HIRSCHMANN et al. PERIDOTITE PARTIAL MELTING III than those enriched in those components; we recall that The differences between batch and fractional (or incremental this effect was observed in our simple binary and ternary batch) melting of fertile peridotite are ilthis model systems (Figs 3 and 5), and an inverse pro- lustrated in Fig. 6. The curves labeled 3 show the portionality between productivity and the concentration calculated characteristics of isobaric, incremental batch of incompatibles in the source in the binary system is melting of MM3 peridotite (with melt removed after apparent from inspection of equations (8) and (9). This 0 1% increments; this is a close approximation to frac- is also the case for more complex model peridotite tional fusion). These curves for fractional fusion can be systems: for example, as shown in Fig. 6a, the MELTScalculated compared with the batch melting curves (labeled 1 in near-solidus productivity for batch melting of Fig. 6) for the same initial bulk composition. As expected the depleted DMM1 peridotite composition is higher from the simple system analogies, there is no difference before exhaustion of cpx at ~8% melting than that between predicted productivities for batch and fractional calculated for batch or for fractional fusion of the fertile fusion right at the solidus. However, just after the first MM3 peridotite composition (although the calculated increment of isobaric fractional melting, the concentrations productivity still increases with increasing F, reflecting of fluxing components in partial melts deproductivity again the decreasing Na content of the melt). Although crease more rapidly with F than in batch melts (Fig. 6b), it is difficult to distinguish in Fig. 6a, productivity for and, as a consequence, temperature rises more rapidly DMM1 is also higher than that of MM3 right at the with F, and isobaric productivity is smaller for fractional solidus, again following the behavior observed in the fusion than for batch fusion (Fig. 6c). This is precisely model binary and ternary systems. It should be noted the difference between fractional and batch melting that that, as also observed for the model ternary, the calculated is generally assumed, but, as already demonstrated, it is temperature required to reach a given melt fraction is a phenomenon particular to near-solidus conditions. After always higher for a depleted composition than for a ~2% melt has been removed, most of the Na (and other, fertile system for the complex peridotite compositions more highly incompatible elements) has been removed (Figs 5c and 6c). Thus, although the melt fraction generated from the system undergoing fractional fusion, so the from fertile peridotite at any given temperature compositions of fractional partial melts, and therefore will always be greater than for a depleted peridotite, the the liquidus temperatures of those partial melts, change difference between the temperature required to generate more slowly with F than in the case of batch partial a particular melt fraction in the depleted composition melting (Fig. 6c). Thus, from ~2% melting to the ex- and that required to generate that same melt fraction is haustion of cpx, the calculated productivity is higher for reduced as melt fraction increases (so long as a phase is fractional melting than it is for batch melting (Fig. 6a), not exhausted from the residue of either source). The just as was observed for the model binary and ternary key point is that owing to these variations in isobaric systems described previously (Figs 3a and 5a). Therefore, productivity, all other things (e.g. the phase assemblage) whereas the common intuition that fractional melting of being equal, the initial adiabatic productivity of any given peridotite is less productive than batch melting is ap- peridotite composition as it upwells past its solidus will plicable near the solidus (but not right at it) and when be inversely correlated with the concentrations of moderately averaged over the entire melting interval, it does not to highly incompatible components (chiefly alkalis apply at higher extents of melting. It should be noted, and volatiles) in the peridotite. however, that even though the productivity for in- Experimental determinations of melt fraction vs tem- cremental batch melting is greater than that for batch perature for fertile peridotite do not show strong evidence melting over a large fraction of the melting interval, at for the near-solidus productivity changes predicted by any given temperature the total extent of melting achieved MELTS or by simple system analysis. The experiments by fractional processes is, as emphasized above, always of Baker & Stolper (1994) and Baker et al. (1995) [T F less than that achieved by batch processes (compare curve relations summarized by Hirschmann et al. (1998b)] and 3 with curve 1 in Fig. 6c). Thus, despite the higher Robinson et al. (1998) both suggest nearly constant pro- productivity of fractional fusion above ~2% melting, ductivity through the lherzolite melting interval. It may according to these calculations the total melt produced be that the real effects in natural peridotite are well by fractional melting never actually catches up to that developed only below melt fractions of ~2 3% [the produced by batch melting. lowest melt fractions explored by the Baker et al. (1995) It is remarkable that the behavior predicted by MELTS experiments] and that the difficulties inherent in using for complex model peridotite compositions matches so variable bulk-composition sandwich experiments to char- well those calculated for the model binary and ternary acterize the temperature melt fraction relations in theoretical systems discussed in the previous sections (compare Figs constant bulk-composition peridotite introduce 3, 5, and 6). This is the case even though the calculated significant uncertainties to the Robinson et al. (1998) melt melting of the MM3 composition involves changing phase fraction estimates. proportions in the residue and changing concentrations 841